Wilson Loops in AdS/CFT

M. Kruczenski

Purdue University

Miami 2012

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Based on work in collaboration with Arkady Tseytlin (Imperial College). To appear.

Summary

●Introduction

String / gauge theory duality (AdS/CFT)

Wilson loops in AdS/CFT = Minimal area surfaces in hyperbolic space

● How to describe in the field theory a string falling into the Poincare horizon

Simple examples: point-like string and short strings.They correspond to inserting a WL in the T-dual theory.

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● Map between short strings and Wilson loops.

The oscillations of a short string (flat space) falling into the horizon are mapped to the oscillations of the world-sheet dual to a Wilson loop.

● Conclusions

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λ small → gauge th.

Gauge theory String theory

λ large → string th.

Strings live in curved space, e.g. AdS5xS5

S5 : Y1

2+Y22+…+Y6

2 = 1

AdS5: X12+X2

2+…-X02-X-1

2 =-1 (hyperbolic space)

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AdS/CFT correspondence (Maldacena)

Gives a precise example of the relation betweenstrings and gauge theory.

Gauge theory

N = 4 SYM SU(N) on R4

Aμ , Φi, Ψa

Operators w/ conf. dim.

String theory

IIB on AdS5xS5

radius RString states w/ E

R

g g R l g Ns YM s YM 2 2 1 4; / ( ) /

N g NYM , 2fixed

λ large → string th.λ small → field th.

Minkowski AdS metric has a horizon at

Hyperbolic space

2d: Lobachevsky plane, Poincare plane/disk

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Poincare coordinates

€

z = ∞

Wilson loops: associated with a closed curve in space.Basic operators in gauge theories. E.g. qq potential.

Simplest example: single, flat, smooth, space-like curve (with constant scalar).

C

C

x

y

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String theory: Wilson loops are computed by finding a minimal area surface (Maldacena, Rey, Yee)

Circle:

circular (~ Lobachevsky plane)

Berenstein Corrado Fischler MaldacenaGross Ooguri, Erickson Semenoff ZaremboDrukker Gross, Pestun

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Maldacena, Rey Yee parallel lines

Drukker Gross Ooguri cusp

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Multiple curves:

Drukker Fiol

concentric circles

Lens shaped: closed with cuspsDrukker Giombi Ricci Trancanelli

New examples using Riemann Theta functions(w/ Sannah Ziama and R. Ishizeki)

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Even more examples: Concentric curves12

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Two main sets of coordinates for hyperbolic space:Poincare and global coordinates. The boundary of global coordinates is RxS3 and the boundary of Poincare is R3,1. The dual field theory lives in the boundary space, therefore string theory in global AdS is dual to gauge theory on RxS3 and in Poincare dual to gauge theory on R3,1. In Minkowski signature, Poincare coordinates have an extremal horizon and, in fact, only cover part of the space.

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Relation between global and Poincare patches

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In this talk we discuss other use for Wilson loops

Consider the computation of a correlation function for the field theory living on S3

Should map to a correlation function of the theory in flat space.

However, Poincare coordinates only cover part of the space and also of the boundary.

A string can ”escape” through the horizon!.

16The disappearing string should be represented by an operator in the field theory. Which operator?

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Consider the case of a small string

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● Alday-Maldacena T-duality

Relates AdS5xS5 to itself, equivalently relates N =4 SYM to itself as a “momentum position” duality.

Proposed by Alday-Maldacena and made more precise by Berkovits-Maldacena and Beisert-Ricci-Tseytlin-Wolf.It involves a novel fermionic form of T-duality.

At the level we need here, it relates two classical solutions in AdS5. The simplest form is in Poincare coordinates and in conformal gauge.

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Metric:

Action:

EOM:

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Simple but useful example:

21A point-like string becomes a straight WL of the T-dual theory.

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Consider now small fluctuations of the string.

Since the string is a small fluctuation of the point-like string we can just study the string in flat space. Moreover, we can use light-cone gauge (since in flat space it is a type of conformal gauge) and the equations reduce to a set of harmonic oscillators.

More formally, start from embedding coordinates:

and expand around X0=1:

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The metric around this (any) point is approximately:

Light-cone gauge:

Most general solution for the other coordinates:

t=0 corresponds to the horizon. Classically we need to specify the shape and velocity of the string on that surface.

r =1..4

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Pictorially:

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There seems to be a problem. These fluctuations should be mapped to fluctuations of the Wilson loop which is in AdS and not in flat space ?!.

Indeed, the action for the fluctuations around Z=s, T=t is (notice interchange of s and )t :

which gives the equations:

?!

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But, in fact, it was already shown by A. Mikhailov that these equations are solved by

So, indeed the equations are like in flat space. In fact using the rules of T-duality we find

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First change to Poincare coordinates

and perform a rescaling (boost) by , e the solution is

T-duality gives:

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The expectation value of the Wilson loop is related to the area of the dual world-sheet

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The energy of the open string ending on the boundary is not conserved (we need do work on the quark to move it along a specified trajectory). The total energy change, however, for these solutions is:

It vanishes from the level matching condition for the closed string. However if we add the left and right movers we get the energy of the closed string:

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The spin is also not conserved.

The velocity and acceleration are evaluated at the boundary. The reason is that the spin defines a conserved current, any increase in spin is due to a flux from the boundary. The total spin absorbed by the open string is precisely equal to the spin of the original closed string.

Conclusions

We review the duality between Wilson loops and minimal area surfaces in hyperbolic space.We then considered the case where a string crosses the Poincare horizon. Using T-duality we found that such string is represented by the insertion of a Wilson loop of the T-dual theory.

In that way we found another interesting application of Wilson loops that can help understand the physics of horizons in the context of AdS/CFT and perhaps more in general.

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